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Fracture Mechanics online class
Fracture Mechanics Online Class
Overview > Linear Elastic Fracture Mechanics (LEFM)
Why do we need fracture mechanics?
You might be wondering why is it necessary to develop a whole theory taking into
account a microscopic defect (the crack) to describe the behavior at the macroscopic
scale. There are two reasons for this:
It is impossible to produce a crack free structure and we have seen that
accounting for the existing defects is required to predict the life of the
structure.
If we apply classical continuum mechanics on a crack problem we are facing
problems of singularities as detailed here below.
Let us analyze the case of a hole in an infinite plate submitted to uniform
traction. Ingis modeled this case by solving the case of an ellipsoidal void (1913),
see Picture I.50.
2a
σmax = σyy (a, 0) = σ∞ (1 +
Picture I.50: Ellipsoidal void in an
infinite plate.
).
(1)
b
The case of a crack can be recovered by taking b → 0. In that case σmax tends to
infinity at the crack tip, meaning that if one wants to limit the stress value to a
threshold, as when designing to remain in the elastic regime, σ∞ needs to tend to 0.
However Griffith and Irwin have shown that a sample with a crack can still sustain a
loading:
−
−−−−−
−−−−−
−
σTS √a ÷ √ E (2γ s + W pl ).
(2)
In order to solve the contradiction between the stress analysis and the experimental observations, fracture mechanics was
developed to answer to the following questions:
For a sample with an existing crack, knowing the crack size, what is the maximum loading leading to crack
propagation?
For a sample with an existing crack submitted to a cyclic loading, if the stress remains lower than the fracture stress,
what is the life of the structure?
In this overview we will first consider the case of linear elastic materials. Note that this assumption will not always hold, but
we will discuss this in details later on.
Linear Elastic Fracture Mechanics (LEFM)
Under this assumption, the laws of mechanics for elastic linear materials can directly be used. In order to solve the problem of
a sample with a crack, Irwin has studied three different loading modes. There are two modes loading the crack in its plane, the
Mode I also called opening mode as it tends to open the crack lips, see Picture I.51, and the Mode II also called sliding mode as it
tends to make the two crack lips slide on each other, see Picture I.52. The Mode III, also called shearing mode, loads the crack
out of its plane, see Picture I.53.
Asymptotic solution
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Picture I.51: Mode I,
opening.
Picture I.52: Mode II,
sliding.
σzz = 0 or ε zz = 0
⎧
⎪
⎪
⎪
⎪
⎪ σyy (θ ± π) = 0
⎨ σxy (θ ± π) = 0
⎧ σxx = σxy = 0
⎪
⎪
⎪
σyy = σzz = 0
.
⎨
⎪ ux = uy = 0
⎪
⎩
⎪
u z (θ) = −u z (−θ)
σzz = 0 or ε zz = 0
⎧
⎪
⎪
⎪
⎪
⎪ σyy (θ ± π) = 0
,
(3)
⎨ σxy (θ ± π) = 0
⎪
⎪
⎪ u x (θ) = u x (−θ)
⎪
⎩
⎪
u y (θ) = −u y (−θ)
Picture I54: Polar
coordinates linked to the
crack tip. The crack is in
red.
Picture I.53: Mode III,
shearing.
,
(4)
⎪
⎪
⎪ u x (θ) = −u x (−θ)
⎪
⎩
⎪
u y (θ) = u y (−θ)
(5)
Classical solid mechanics can be applied on each of these modes in order to compute the stress
and strain field. To do so the polar coordinates linked to the crack tip will be used, see Picture
I.54. The only things that differentiate the three modes are the boundary conditions of the
problem, which can directly be written in terms of the polar coordinates as (3-5). Once the
boundary conditions are known, it is possible to solve the problem. The calculations will be
presented in details in the next lecture, but we can already discuss the results. The solution is
obtained as series in √r, where r is the distance to the crack tip, see Picture I.54, and reads for
Mode I:
⎧
⎪ σxx =
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪ σyy =
⎪
⎪
⎪
⎨
C
cos
√r
θ
[1 − sin
2
C
3θ
[1 + sin
C
θ
2
3θ
θ
0
] + O (r )
sin
2
√r
θ
2
3θ
θ
cos
sin
2
θ
where κ is an expression of the Poisson ratio ν and is equal to
state. For Mode II one can find a similar solution:
3−ν
1+ν
,
0
σxy =
cos
cos
sin
+ O (r )
⎪
2
2
2
√r
⎪
⎪
⎪
C(1+ν)
⎪
θ
2
⎪
⎪ ux =
) [κ − 1 + 2 sin (
√r cos (
⎪
E
2
⎪
⎪
⎪
⎪
C(1+ν)
⎪
θ
2
⎩
⎪u =
) [κ + 1 − 2 cos (
√r sin (
y
E
0
] + O (r )
2
2
θ
2
θ
2
)] + O (r)
)] + O (r)
for the plane stress state and to 3 − 4ν for the plane strain
C
3θ
θ
θ
0
⎧
sin
[2 + cos
cos
] + O (r )
⎪ σxx = −
⎪
2
2
2
√r
⎪
⎪
⎪
C
3θ
⎪
θ
θ
0
⎪σ
cos
sin
cos
+ O (r )
⎪ yy =
⎪
2
2
2
√r
⎪
⎪
⎪
⎪
⎪
C
3θ
θ
θ
0
cos
[1 − sin
sin
] + O (r )
.
⎨ σxy =
2
2
2
√r
⎪
⎪
⎪
⎪
C(1+ν)
⎪
θ
θ
⎪
2
⎪ ux =
) [κ + 1 + 2 cos (
)] + O (r)
√r sin (
⎪
E
2
2
⎪
⎪
⎪
⎪
C(1+ν)
⎪
θ
θ
2
⎩
⎪
uy = −
) [κ − 1 − 2 sin (
)] + O (r)
√r cos (
E
(6)
2
(7)
2
Finally, for Mode III the solution reads:
C
θ
0
⎧ σxz = −
sin
+ O (r )
⎪
2
√r
⎪
⎪
⎪
⎪
⎪
C
θ
0
cos
+ O (r )
.
⎨ σyz =
2
r
√
⎪
⎪
⎪
⎪
4C(1+ν)
⎪
θ
⎩
⎪u =
) + O (r)
√r sin (
z
E
(8)
2
Let us first consider a mode I loading and let us analyze the dominant term of
σyy
(6), which is the stress field characterizing
the stress concentration of a plate under tension, see Picture I.50. Clearly the dominant term near the crack tip is in
C
√r
,
meaning there is a singularity at the crack tip. Thus the value of the stress cannot be used to determine whether the crack will
propagate or not. The idea of Irwin was thus to consider "how fast the stress tends to infinity" near the crack tip. To do so he
has defined the stress intensity factor (SIF) to circumvent the singularity of the solution:
−
−− mode I
−−
|
) = C √2π
⎧ KI = limr→0 (√2πr σyy
θ=0
⎪
−
−− mode II
−−
|
) = C √2π
⎨ KI I = limr→0 (√2πr σxy
⎩
⎪
−
−− mode
KI I I = limr→0 (√2πr σyz
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θ=0
III
|
θ=0
.
(9)
−−
) = C √2π
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In this equation we have also treated the two other fracture modes by considering the dominant stress terms of mode II (7) and
−
of mode III (8). For a given loading mode, this SIF, expressed in MPa ⋅ √−
m , characterizes the stress evolution near the crack
tip:
mode
⎧
⎪
σ
⎨
⎩
⎪
i
mode i
u
=
Ki
√ 2πr
f
mode i
(θ)
,
−−
r
mode i
= Ki √
g
(θ)
(10)
2π
where f and g are functions defined for each mode but independent of the loading and geometry (as long as we consider the
asymptotic value). The loading and geometry effects are thus fully reported to the value of the SIF. Irwin has thus the idea to
consider the value of the SIF to detect the crack propagation. Indeed experiments have shown that for a given material, which
obeys to the LEFM assumption, the crack propagates if the SIF reaches a threshold KC called the toughness of the material.
Crack propagation criterion and toughness
From the previous section, one can write the crack propagation criterion in mode I as:
KI
{
< KI C →
the crack does not propagate,
≥ KI C →
the crack does propagate,
(11)
where KI C is the mode I toughness. Remember, under the LEFM assumption:
KI
Picture I.55: Typical values of the
toughness and yield stress for
different materials (ranges provided
for illustration purpose only).
depends on the geometry and loading conditions only,
depends on the material only.
KI C
−
Orders of magnitude for the toughness are for example for concrete 0.2-1.4 MPa ⋅ √−
m
and typical values for other materials are reported in Picture I.55. Materials for which
−
the toughness is lower than 30 to 40 MPa ⋅ √−
m are considered as brittle materials.
Picture I.56: Evolution of the
toughness in function of the
temperature for steel (ranges
provided for illustration purpose
only).
Ductile materials have a higher toughness, but as we will discuss it in length later they
usually do not satisfy to the LEFM assumption as their behavior is no longer elastic. As
discussed before, some materials have a brittle behavior at low temperature and a
ductile behavior at high temperature. For such materials the toughness depends on the
temperature and there exists a transition region as shown in Picture I.56. Note that at
high stress rate these materials also loose ductility.
Evaluation of the SIFs
The next question which arises is "How do we can determine this SIF" that has to be compared to the material toughness. As
said, this SIF depends on the geometry and on the loading condition. In practice, there exist 5 different methods.
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Picture I.57: Centered crack in an
infinite plate under mode I loading.
Fracture Mechanics online class
Picture I.58: Centered
crack in an infinite plate
under mode II loading.
Picture I.59: Centered
crack in an infinite plate
under mode III loading.
Analytical method. This can only be achieved for simple cases as for a crack of size 2a in an infinite plate as shown in
Picture I.57, I.58 and I.59. In these cases the following analytical results for the different SIFs can be computed as it will
be done in the next lecture:
−−
KI = σ∞ √πa
⎧
⎪
−−
⎨ KI I = τ∞ √πa .
⎩
⎪
−−
KI I I = τ∞ √πa
(12)
For more complicated problems, the SIFs cannot be computed in a closed-form, but they follow
−−
KI = βI σ∞ √πa
⎧
⎪
−−
,
⎨ KI I = βI I τ∞ √πa
⎩
⎪
−−
KI I I = βI I I τ∞ √πa
(13)
where the βi are now values that depend on the geometry and on the crack size.
Superposition principle. As the problem is fully linear, we can apply the superposition principle. Note that one can add
the SIFs related to the same mode only. Indeed according to (10 ), the functions f and g, which depend on θ, are different
for the different modes. Thus adding the stress tensor (by superposition principle) does not correspond to adding the SIFs
of different modes. Thus for two different loadings a and b we have Ki = Kia + Kib , but for two different modes, we
have K ≠ KI + KI I !
Use of handbooks. Solutions for different (simple) geometries obtained by different ways are tabulated in handbooks.
Finite Element (FE) method. We will extensively describe how to determine the SIFs from a FE resolution.
Energetic approach. This method makes the bridge between the work of Griffith based on the fracture energy and the
asymptotic solution. We will introduce this point briefly here below, but a complete lecture is dedicated to it.
Measurement of the toughness
Once the SIFs of a structure are computed, they have to be compared to the toughness
of the material. This value has to be measured experimentally by strictly following the
procedure defined by the ASTM E399 Standard. Different kinds of samples can be
manufactured from a given material as the Single Edge Notched Bend (SENB) specimen
as shown in Picture I.60. To remain conservative, it is necessary to be in the plane
strain state (P-ε ), which means that the specimen is thick enough (a quantity yet to be
defined). Indeed in the plane stress state (P-σ) the measured toughness is higher as it
will be explained later on. The specimen has initially a V-notch, which will initiate the
crack. The evaluation of the toughness requires the following steps
Picture I.60: SENB test and
notations.
Calibrate the measurement technics of the three-point bending test to measure
the evolution of the displacement δ and of the crack-mouth opening ν in terms
of the loading force P .
Initiate an initial crack by fatigue by applying a cyclic loading well below the
fracture loading.
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Measure the compliance
ν
P
of
of the cracked specimen and determine the
δ
P
value a of the crack using FE predictions
Picture I.61: Determination of the
critical loading P . Left: using the
95%-offset. Right: Maximum loading
value if reached before the 95%offset.
ν
P
= f (a)
.
Perform the three-point bending test to measure the critical loading PC . This
value is obtained using the experimental P-ν loading curve, see Picture I.61. PC
is defined as the loading value obtained for a 95%-offset of the initial structural
stiffness. However if the maximum value of P is reached before this offset, this
corresponds to the critical value PC .
The toughness is then deduced using the expression of the SIF obtained from FE
simulations:
c
PL
KI =
tW
where
a
W
a
3
f (
),
(14)
W
2
depends on the test specimen (SENB ...). Substituting
in (14 )
PC
directly gives KI C .
Relation with the fracture energy
In the previous subsection we have seen how to determine whether a crack will
propagate or not. However in the last section we have seen that the critical crack size
is related to the fracture energy. This topic will be extended in a coming lecture, but we
can already get some insight of the phenomenon by considering a Mode I loading of a
2a -long centered crack in an infinite plate. Considering the polar coordinates at one of
the two crack tips (before crack propagation), see Picture I.62, the stress and
displacement fields ahead of the crack tip are deduced from (6-12 ) by considering
θ = 0 and x = r + a , leading to:
σ∞ √ a
⎧
⎪ σ 0 (θ = 0, r = x − a) =
⎨
⎩
⎪
Picture I.62: Initial crack size and
associated polar coordinates.
√2(x−a)
0
uy
.
(15)
(θ = 0, r = x − a) = 0
Now we consider that the crack has grown straight ahead to reach a size 2(a + Δa),
see Picture I.63. The part of the crack lips which now corresponds to the increment Δa
was before a sound material with the stress and displacement fields (15 ). This part
corresponds now to opened crack lips, and the displacement and stress fields of the
upper/lower crack lip can still be deduced from (6-12 ), but this by considering
θ → ±π and x = a + Δa − r , leading to:
1
Picture I.63: Crack has propagated
to reach a size 2(a + Δa) , and
associated polar coordinates.
⎧ σyy (θ = π, r = a + Δa − x)
⎪
⎪
= 0
⎨ u1 (θ = ±π, r = a + Δa − x)
y
= ±
⎪
⎩
⎪
σ∞ (1+ν)(κ+1)
.
(16)
E√ 2
−
−
−−
−
− −
−−−−−−
−−
√a + Δa √a + Δa − x
From (15 ) and (16 ), we can see that the part of interest is subjected to a stress field
0
σyy
but at zero-displacement before the
crack propagation, while it is subjected to an opening displacement
but is stress-free after crack propagation. There is thus
an energy released by the structure as the stress tensor exhibits a work during the crack propagation. This energy can be
computed as:
1
uy
1
a+Δa
ΔE int = −4dz ∫
uy
∫
σyy duy dx,
(17)
0
uy
a
where the factor 4 comes from the fact that there is double symmetry around the x and y axes (Remembering that there is a
crack increment of size Δa on each side of the centered crack). As we consider a linear elastic material, during the crack
opening σyy has to decrease linearly with the displacement uy , and the integral over the displacement can be easily be solved
leading to:
a+Δa
ΔE int = −2dz ∫
0
1
σyy uy dx.
(18)
a
Using the values (15 -16 ), and applying the change of variables x
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= a + Δa cos
2
θ
, the energy released reads:
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2
dzσ∞
−
−
−
−
−−
−
−−
√ a (a + Δa) (1 + ν ) (1 + κ)
ΔE int = −
πΔa.
(19)
2E
If we relate this energy to the increment of crack surface dA = 2Δadz and change the sign as to have a positive value, we
have an important concept for fracture mechanics, which is the energy release rate G :
dE int
G = −
.
(20)
dA
This energy corresponds to the energy released by the structure by unit increment of the crack surface. It reads in our case:
G = −
dE int
ΔE int
= − lim
dA
Δa→0
=
πaσ∞2
(1 + ν ) (κ + 1) =
4E
2Δadz
πaσ∞2
E
′
K
=
2
I
E
′
,
(21)
where
E
′
=
E
4E
= {
Plane stress state
.
E
(1 + ν ) (κ + 1)
1−ν
(22)
Plane strain state
2
In this example we have assumed a crack lying in an infinite plate. In the next lectures, it will be showed that for a crack
propagating straight ahead under mode I we always have under the LEFM assumption
K
2
I
G =
E
′
(23)
.
We will also see later than as the energies can be added, we get for the global energy release rate, not forgetting the assumption
of a straight forward growing crack:
K
2
I
G =
E
′
K
2
(1 + ν ) K
II
+
E
+
′
2
III
(24)
E
This expression relates the SIFs to the energy that the structure releases when a crack propagates. This is a first link between
the energy and the stress approaches. Going further on with the energy approach, we can follow Griffith (1920) approach. The
total energy of the specimen can be seen as the sum of the internal (elastic) energy Eint with the energy Γ needed to create a
crack of surface A :
E = E int + Γ
(25)
Deriving this equation with respect to the crack surface A and considering that the total energy is conserved (as no dissipation
is considered) yields:
∂E
=
∂ E int
∂A
∂Γ
+
∂A
= 0.
(26)
∂A
Using the definition of the energy release rate (20 ), this last equation leads to
G = −
dE int
dΓ
=
dA
Assuming the crack is actually growing, the term
dΓ
dA
.
(27)
dA
actually corresponds to twice the surface energy
γs
by definition of the
surface energy (and as we form two surfaces A ). Thus the former equation reads
G = −
dE int
dA
dΓ
=
= 2γ s = G c .
(28)
dA
The meaning of this equation is the following. For a crack to grow the energy that the structure would have to realease due to
the crack growth (by unit crack surface) have to be larger or equal to G C the energy required to form a crack (of unit surface).
For brittle material this is twice the surface energy and corresponds to the energy required to have cleavage. As we said that the
crack would grow if KI ≥ KI C , the crack would grow if G ≥ G c .
Assuming the crack does propagate, then combining (21 ), in which we substitute σ∞ by the strength of the material σTS , with
(28 ) leads to
−
−
−
−−
2γ s E
σTS = √
,
(29)
πa
which is nothing else than the Griffith formula for brittle materials seen before.
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Let us now consider the practical implication of this formula:
For glass (2γ s
=
2 J / m2 and E
=
−
60 GPa): σTS √a
1
= 195kPa m
2
For mild steel at room temperature (ductile plastic work will be dominating,
−
σTS √a = 115MPa m
200 kJ / m2 and
2γ s =
E =
210 GPa):
1
2
This example shows how ductile materials have a much higher tensile strength for the same crack size, explaining the troubles
of the liberty ships.
Direction of crack propagation
We have determined whether a crack grows when loaded in mode I. In that case, by
symmetry the crack propagates straight ahead. What would happen when the loading
mode is mixed (for example a combination of mode I and mode II)?
What is the condition for crack growth?
In which direction would it grow?
To answer to these questions, let us consider the mixed mode loading of Picture I.64, in
which the loading direction is oriented with an angle β ∗ with respect to the crack (
π
∗
β =
corresponding to a pure mode I loading). As it will be demonstrated later, the
2
SIFs of the two modes can be related from:
cot β
⋆
=
KI I
.
(30)
KI
Picture I.64: Mixed loading mode.
One assumption to define the direction of the crack growth is to state that the crack
grows in the direction so that the SIF related to Mode I in the new frame is maximum.
This actually corresponds to maximizing σyy in this new frame, i.e. it corresponds to
maximizing the hoop stress σθθ in the orginal polar coordinates. This latter can be
computed as (this will be demonstrated later):
σθθ =
KI
−
−−
√2πr
[cos
3
θ
−
2
3 cot β⋆
θ
sin θ cos
2
].
(31)
2
The direction θ⋆ of the maximum hoop stress is thus given by:
Picture 65: Evolution of the hoop
stress σ with respect to θ for
different mixed loading conditions
characterized by β .
∂ θ σσσ |
θ
θθ
∗
⋆
= 0,
(32)
and the crack growth criterion (which is thus mode I in that direction) becomes
−
−−
⋆
(√2πr σθθ (r, θ )) ≥ KI C .
(33)
Picture I.65 shows the evolution of the stress σθθ (31 ) with respect to the angle θ and different mixed loading cases β ∗ . The
direction of the maximum hoop stress is also reported as a red curve. What can be seen is that for a positive loading angle, the
crack will kink with a negative angle.
J-integral
As explained previously the concept of crack propagation is strongly linked to the
energy released during crack propagation. One practical way to compute the energy,
without requiring the crack to actually grow, is the J-integral introduced by Rice. The Jintegral corresponds to the energy flowing through a contour embedding the crack tip
as shown in Picture I.64, and reads
J = ∫
[U (ε)nx − ∂ x u ⋅ T] dl,
(34)
Γ
in which the first term corresponds to the flow parallel to the crack of the material
energy and the second term to the flow parallel to the crack of the traction energy on
the contour (as T = σ ⋅ n and as n is the outward unit normal of the contour).
Picture I.66: Contour of the Jintegral.
The J-integral has the following properties:
It is path independent if the contour Γ embeds a straight crack tip;
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There is no assumption on how the crack will grow;
If the crack grows straight ahead, J = G;
If the sample is linear elastic, J
K
=
E
2
I
′
K
+
2
II
E
′
K
+
2
III
2μ
;
The J integral can be extended to plasticity when unloading is prohibited.
Because of these properties, the J-integral is:
An efficient numerical tool to evaluate the SIFs in FE simulations;
Useful for non-perfectly brittle materials.
We will expend the concept of the J-integral in a coming lecture.
Case of ductile materials
In the case of ductile materials, two more aspects need to be considered.
Resistance curve and crack stability
We have seen in (28 ) that a crack propagates if G ≥ G C . For a perfectly brittle
material the fracture energy is constant and equal to 2γ s . For other materials there
will be a plastic flow at the crack tip as a stress cannot physically reach the infinity as
predicted by the LEFM approach. This plastic flow arises in the active plastic zone as
illustrated in Picture I.67. Evidently this active plastic zone moves with the crack tip in
case of crack growth, meaning that the crack is opening in a zone affected by
permanent plastic deformations: the plastic wake. This makes the crack more difficult
to be opened, which corresponds to an increase of the apparent fracture energy G C
with the crack propagation Δa . This is call the resistance and G c is replaced by
Rc (a). Some material can have a steady-state in the resistance curve as in Picture I.67,
other not. Note that the curve does not only depend on the material, but also on the
geometry as the thickness.
Picture I.67: Resistance of a nonperfectly brittle material.
Picture I.68: Crack stability for nonperfectly brittle materials.
in this case G is always larger than RC
A crack is stable if G
= Rc (a)
and
dG
da
During the crack propagation the geometry of the structure changes (as the crack
changes) so does the energy release rate G . Assuming a constant force loading, G
increases with the crack length at it will be shown later. On the one hand, for a
perfectly brittle material as G C is constant, once the crack initiates it keeps
propagating as G remains larger than the constant G C . On the other hand, for a
ductile material Rc also evolves with the crack size and a crack can now be stable.
This is illustrated in Picture I.68. Assuming an initial crack size a0 , the crack
propagates if the loading reach σ = σ2 . However under this loading condition, for a
crack size a0 + Δa, G (σ2 , a0 + Δa) < RC (a0 + Δa) and the crack stops. To make
the crack growth one has to increase the loading up to σ3 but the crack would also
eventually stop: the crack is stable. This behavior changes if the loading reaches σ4 as
for any crack size a : the crack is unstable. Mathematically, this reads:
≤
dR c
da
, or unstable if G
= Rc (a)
and
dG
da
>
dR c
da
.
LEFM limit
We have just considered ductile materials, which exhibit plastic deformations.
However up to now we have considered the LEFM approach which assumes elastic
linear materials. The validity of the method is thus questionable.
Let us first assume there are no plastic deformations. The LEFM asymptotic method
thus predicts a stress field in
1
√r
tending to infinity at the crack tip, see Picture I.69.
This is of course unphysical and some irreversible behaviors occur near the crack tip
(dark green zone on Picture I.69). The asymptotic solution is thus not valid in this zone.
Now faraway of the crack tip, the asymptotic solution has neglected the terms in r 0 ,
√r , r ... and does not capture the structural response (faraway from the crack tip the
asymptotic solution predicts a stress tending to zero, which is not usually the correct answer), see Picture I.69. As a summary,
the LEFM solution is not valid faraway from the crack tip and is not valid close to the crack tip in a zone whose size depends on
the degree of plastic deformations involved.
Picture I.69: Validity region of the
asymptotic LEFM solution.
For some materials, considered as brittle, if the sample is large enough the dark green zone of Picture I.69 remains small
enough and there exists a cylindrical zone (light green in Picture I.69), in which the asymptotic solution if valid. In this
case the LEFM approach can be used.
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Fracture Mechanics online class
If the material is more ductile, or if the sample is too small, the dark green region becomes bigger with respect to the
structural response stress field zone and the region of validity of the asymptotic solution vanishes. LEFM approach
cannot be used anymore.
In coming lectures we will see how to validate the applicability of the LEFM approach and also consider Non-Linear Fracture
Mechanics (NLFM) when required.
If the crack front plastic zone
rp
remains smaller but is of comparable order of
magnitude with respect to the crack length a , see Picture I.70, the LEFM approximation
still holds (i.e. there exists a region of validity as in Picture I.69) and we are typically
under the Small Scale Yielding assumption (SSY). A first order correction can be
achieved to improve the accuracy of the method: when determining the SIFs one needs
to replace the crack length by the effective crack length aeff , which accounts for the
blunting in the plastic zone, see Picture I.70. The system of equations to be solved
reads:
rp
Picture I.70: Small Scale Yielding
assumption (SSY).
a → a eff = a +
(35)
3
πK
rp =
2
I
(a eff )
2
(36)
8σp
When the plastic zone becomes too large, when the applied stress is larger than half the yield stress of the material, none of the
discussed approximations hold and it is necessary to use the NLFM approach.
Ludovic Noels, E-mail: L.Noels @ ulg.ac.be • University of Liège • Computational & Multiscale Mechanics of Materials (CM3)
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